326 ANALYTICAL MECHANICS
eliminated by taking the origin as the position of zero potential energy. Thus we have
E7=/32g2+/33<Z3+ • • • • (XXVII')
But since the vibrations are supposed to be small, q remains a small quantity during the motion. Therefore the higher powers of q are negligible compared with q2. Thus neglecting the higher terms we obtain the following expression foi the potential energy of the system.
[7=|/3g2, (XXVIII)
where i j8 = ft-
The kinetic energy, on the other hand, takes the form
T=%aq\ (XXIX)
where a is a constant and q = -5* . But since the system is
Cut
conservative the sum of its dynamical energy remains constant. Therefore
E= T+U = i *g* + I ft*. (XXX)
Differentiating both sides of the last equation with respect
to the time,
aq + pq= 0, (XXXI)
which is the differential equation of simple harmonic motion. Therefore we have
q = a sin \ (t + fc) (XXXII)
!La
and p=27r\/~~. (XXXIII)
Hence the main part of Lagrange's method consists of selecting the coordinate which defines the position of the system in such a way as to make the expressions for the kinetic and potential energies of the forms